Poisson-Nernst-Planck Research Articles

Model reduction-based initialization methods for solving the Poisson-Nernst-Plank equations in three-dimensional ion channel simulations

The Poisson-Nernst-Planck (PNP) model is frequently-used in simulating ion transport through ion channel systems. Due to the nonlinearity and coupling of PNP model, choosing appropriate initial values is crucial to obtaining a convergent result or accelerating the convergence rate of finite element solution, especially for large channel systems. Continuation is an effective and commonly adopted strategy to provide good initial guesses for the solution procedure. However, this method needs multiple times to solve the whole system at different conditions. We utilize a reduced model describing a near-or partial-equilibrium state as an approximation of the original PNP system (describing a non-equilibrium process in general). Based on the reduced model, we design three initialization methods for the solution of PNP equations under general conditions. These methods provide the initial guess of the PNP system by solving a specifically designed Poisson-Boltzmann-like model, Smoluchowski-Poisson-Boltzmann-like model, and linear approximation model. Simulations of potassium channels 1BL8 and 2JK4 demonstrate that these methods can effectively reduce the number of Gummel iteration steps and the total CPU time in the solution of the PNP equations, and especially do not need the continuation approach anymore. The reason is that these initial guesses can approximate the PNP solution well in the channel region. Besides, our numerical experiments demonstrate that as one of the initialization methods, the linear approximation method can even produce very close results such as current-voltage curves to that from the PNP model when the membrane potential is not high.

Asymptotic analysis of compression sensing in ionic polymer metal composites: The role of interphase regions with variable properties

Ionic Polymer Metal Composites (IPMCs) consist of two noble metal electrodes plating an electroactive polymeric membrane, referred to as ionomer, which is electroneutralised by a solvent including mobile ions. The IPMC manufacturing leads to thin interphase regions next to the electrodes, the so-called Composite Layers (CLs), in which metal atoms occupy interstitial sites within the ionomer. In this work we extend previous efforts of our group on IPMC compression sensing to include the important effect of CLs, where large variations of the electrochemical properties occur. In IPMC compression sensing the application of a through-the-thickness displacement leads to a shortcircuit electric response, here assumed to be governed by a linearised <i>modified</i> Poisson-Nernst-Planck (PNP) system of partial differential equations (PDEs), to be solved for the time-evolving electric potential and mobile ions concentration as functions of the displacement field evaluated through the linear momentum balance. The variation of material properties in the CLs requires the simultaneous integration of the governing system of PDEs in three regions: the membrane and the two CLs. To this purpose, we resort to the perturbative method of matched asymptotic expansions. Except for a numerical inverse Laplace transform, this allows us to obtain an analytical solution through which we establish an equivalent circuit model elucidating the main features of the IPMC sensing behaviour. We validate and discuss the analytical solution through comparison with finite element analyses, whereby we also numerically solve the nonlinear modified PNP systems fully coupled with the linear momentum balance accounting for the electrochemical stresses. We finally provide some insight into the role of CLs in the IPMC sensing behaviour, by assessing its sensitivity to some key parameters. We expect the obtained results to aid the design of optimised IPMC sensors.

Electrokinetic viscous rotating disk flow of Poisson-Nernst-Planck equation for ion transport

The main features of the present numerical model is to explore the behavior of an ionic transport electroviscous boundary layer flow over a rotating disk. For this purpose, the Nernst-Planck equation and Poisson's equation together with the traditional Navier Stokes equations are simulated for the conservation of ionic species. The Poisson-Nernst-Planck (PNP) equation neglects sterile effects and ion-ion interactions, which is widely recognized by the electro-chemists community, leading to advancement of various mathematical models. The modeled governing equations of the fluid flow are transformed to dimensionless ordinary differential equations under Von Karman's approach. The Parametric Continuation Method (PCM) is applied, in order to analyzed the numerical simulation of the problem. For validity of the method the results are compared with another numerical method (bvp4c) and some previous published work, seem to be in a very good agreement to each other. It is found that the potential gradient becomes smaller as Debye length parameter K increases and exposing more counter-ions to the flow which results an increase in the fluid velocity. The physical constraints impact on radial, axial and tangential velocity and on both positive and negative charge profile are sketched and briefly discussed.

Structure-preserving and efficient numerical methods for ion transport

Ion transport, often described by the Poisson–Nernst–Planck (PNP) equations, is ubiquitous in electrochemical devices and many biological processes of significance. In this work, we develop conservative, positivity-preserving, energy dissipating, and implicit finite difference schemes for solving the multi-dimensional PNP equations. A central-differencing discretization based on harmonic-mean approximations is employed for the Nernst–Planck (NP) equations. The backward Euler discretization in time is employed to derive a fully implicit nonlinear system, which is efficiently solved by a newly proposed Newton's method. The improved computational efficiency of the Newton's method originates from the usage of the electrostatic potential as the iteration variable, rather than the unknowns of the nonlinear system that involves both the potential and concentration of multiple ionic species. Numerical analysis proves that the numerical schemes respect three desired analytical properties (conservation, positivity preserving, and energy dissipation) fully discretely. Based on advantages brought by the harmonic-mean approximations, we are able to establish estimate on the upper bound of condition numbers of coefficient matrices in linear systems that are solved iteratively. The solvability and stability of the linearized problem in the Newton's method are rigorously established as well. Numerical tests are performed to confirm the anticipated numerical accuracy, computational efficiency, and structure-preserving properties of the developed schemes. Adaptive time stepping is implemented for further efficiency improvement. Finally, the proposed numerical approaches are applied to characterize ion transport subject to a sinusoidal applied potential.

Electric discharge of electrocytes: Modelling, analysis and simulation

In this paper, we investigate the electric discharge of electrocytes by extending our previous work on the generation of electric potential. We first give a complete formulation of a single cell unit consisting of an electrocyte and a resistor, based on a Poisson-Nernst-Planck (PNP) system with various membrane currents as interfacial conditions for the electrocyte and a Maxwell’s model for the resistor. Our previous work can be treated as a special case with an infinite resistor (or open circuit). Using asymptotic analysis, we simplify our PNP system and reduce it to an ordinary differential equation (ODE) based model. Unlike the case of an infinite resistor, our numerical simulations of the new model reveal several distinct features. A finite current is generated, which leads to non-constant electric potentials in the bulk of intracellular and extracellular regions. Furthermore, the current induces an additional action potential (AP) at the non-innervated membrane, contrary to the case of an open circuit where an AP is generated only at the innervated membrane. The voltage drop inside the electrocyte is caused by an internal resistance due to mobile ions. We show that our single cell model can be used as the basis for a system with stacked electrocytes and the total current during the discharge of an electric eel can be estimated by using our model.

Ambipolar diffusion in the low frequency impedance response of electrolytic cells

The electrical response of an electrolytic cell to an external ac excitation is analysed by solving the equations of the Poisson–Nernst–Planck (PNP) continuum model for two ions (ambipolar) and one mobile ion diffusive systems. The theoretical predictions of the ambipolar system, formed by positive ions of mobility μp and negative ions of mobility μm, are investigated in the limit in which one of the mobilities goes to zero. The analysis reveals that these predictions correspond to the ones arising from the one mobile ion diffusive system only in the frequency range ω ≫ ωDμm/μp, in which ωD is the Debye’s frequency. For very low frequencies, it shows that the physical system formed by two mobile ions, one of which has a very low mobility, is clearly distinct from the physical system in which just one of the ions is mobile. We argue that apparent deviations of the experimental spectra from the predictions of the PNP model in the low frequency region, usually interpreted as an interfacial property, may be connected with the difference in the diffusion coefficients of cations and anions.

A fully discrete positivity-preserving and energy-dissipative finite difference scheme for Poisson–Nernst–Planck equations

The Poisson–Nernst–Planck (PNP) equations is a macroscopic model widely used to describe the dynamics of ion transport in ion channels. In this paper, we introduce a semi-implicit finite difference scheme for the PNP equations in a bounded domain. A general boundary condition for the Poisson equation is considered. The fully discrete scheme is shown to satisfy the following properties: mass conservation, unconditional positivity, and energy dissipation (hence preserves the steady state). Solvability of the semi-discrete scheme is proved and a simple fixed point iteration is proposed to solve the fully discrete scheme. Numerical examples in both 1D and 2D and for multiple species are presented to demonstrate the convergence and properties of the proposed scheme.

Superconvergent estimate of a Galerkin finite element method for nonlinear Poisson–Nernst–Planck equations

A Galerkin finite element method (FEM) is developed for nonlinear Poisson–Nernst–Planck (PNP) equations. Based on the combination technique of the element’s interpolation and Ritz projection together with the special mean value approach, the superclose and superconvergence estimates with order O(h2) are derived under weaker regularity requirements of the exact solution. Some numerical results are provided to confirm the theoretical analysis.

Existence of Approximate Solutions for Modified Poisson Nernst-Planck Describing Ion Flow in Cell Membranes

Dynamics of ions in biological ion channels has been classically analyzed using several types of Poisson-Nernst Planck (PNP) equations. However, due to complex interaction between individual ions and ions with the channel walls, minimal incorporation of these interaction factors in the models to describe the flow phenomena accurately has been done. In this paper, we aim at formulating a modified PNP equation which constitutes finite size effects to capture ions interactions in the channel using Lennard Jonnes (LJ) potential theory. Particularly, the study examines existence and uniqueness of the approximate analytical solutions of the mPNP equations, First, by obtaining the priori energy estimate and providing solution bounds, and finally constructing the approximate solutions and establishing its convergence in a finite dimensional subspace in L2, the approximate solution of the linearized mPNP equations was found to converge to the analytical solution, hence proof of existence.

Deconvolution of electroosmotic flow in hysteresis ion transport through single asymmetric nanopipettes.

Unveiling the contributions of electroosmotic flow (EOF) in the electrokinetic transport through structurally-defined nanoscale pores and channels is challenging but fundamentally significant because of the broad relevance of charge transport in energy conversion, desalination and analyte mixing, micro and nano-fluidics, single entity analysis, capillary electrophoresis etc. This report establishes a universal method to diagnose and deconvolute EOF in the nanoscale transport processes through current-potential measurements and analysis without simulation. By solving Poisson, Nernst-Planck (PNP) with and without Navier-Stokes (NS) equations, the impacts of EOF on the time-dependent ion transport through asymmetric nanopores are unequivocally revealed. A sigmoidal shape in the I-V curves indicate the EOF impacts which further deviate from the well-known non-linear rectified transport features. Two conductance signatures, an absolute change in conductance and a 'normalized' one relative to ion migration, are proposed as EOF impact (factor). The EOF impacts can be directly elucidated from current-potential experimental results from the two analytical parameters without simulation. The EOF impact is found more significant in intermediate ionic strength, and potential and pore size dependent. The less-intuitive ionic strength and size dependence is explained by the combined effects of electrostatic screening and non-homogeneous charge distribution/transport at nanoscale interface. The time-dependent conductivity and optical imaging experiments using single nanopipettes validate the proposed method which is applicable to other channel type nanodevices and membranes. The generalizable approach eliminates the need of simulation/fitting of specific experiments and offers previously inaccessible insights into the nanoscale EOF impacts under various experimental conditions for the improvement of separation, energy conversions, high spatial and temporal control in single entity sensing/manipulation, and other related applications.

Current Blockage of PSA molecular in Si3N4/Si/Si3N4 Sandwich Nanopore

Abstract The paper investigates the current characteristics of prostate specific antigen (PSA) in sandwich nanopore through finite element simulation. By coupling Poisson-Nernst-Planck (PNP) and modified Navier-Stokes (NS) equations, the current blockages during the modification of Si3N4/Si/Si3N4 sandwich nanopore on the silicon surface and the accurate capture of PSA tumor markers in nanopore were calculated. The influence of the size of nanopore on current blockage was explored. The result shown that the diameter of nanopore has more influence than the length of nanopore on the current blockage.

A perturbation solution to the full Poisson-Nernst-Planck equations yields an asymmetric rectified electric field.

We derive a perturbation solution to the one-dimensional Poisson-Nernst-Planck (PNP) equations between parallel electrodes under oscillatory polarization for arbitrary ionic mobilities and valences. Treating the applied potential as the perturbation parameter, we show that the second-order solution yields a nonzero time-average electric field at large distances from the electrodes, corroborating the recent discovery of Asymmetric Rectified Electric Fields (AREFs) via numerical solution to the full nonlinear PNP equations [Hashemi Amrei et al., Phys. Rev. Lett., 2018, 121, 185504]. Importantly, the first-order solution is analytic, while the second-order AREF is semi-analytic and obtained by numerically solving a single linear ordinary differential equation, obviating the need for full numerical solutions to the PNP equations. We demonstrate that at sufficiently high frequencies and electrode spacings the semi-analytical AREF accurately captures both the complicated shape and the magnitude of the AREF, even at large applied potentials.

Microscopic insights into the ion transport in graphene-based membranes with different interlayer spacing distributions

The ion transport in nanoscale pores/channels are totally different to that in the macroscopic level. Here, a combined method based on the Poisson-Nernst-Planck (PNP) and Navier–Stokes (NS) equations is applied to study the ion transport in graphene oxide membranes (GOMs) with different interlayer spacing distributions by controlling the offset angle of the middle nanosheet of the model. The layered membrane with suitable interlayer spacing distribution has strong electrical double layer (EDL) overlapping and relatively weak spatial confinement, which improves the ion selectivity and water permeation. This phenomenon can be further understood by analyzing microscopic properties including the local density and flux profiles with considering the effects of surface charge and bulk concentration. This work could provide macroscopic insights into the ionic flow in laminar GOMs at a different angle and possible improvement of the GOMs performances.

Solution of Ion Channel Flow Using Immersed Boundary-Lattice Boltzmann Methods.

Poisson-Nernst-Planck (PNP) model has been extensively used for the study of channel flow under the influence of electrochemical gradients. PNP theory is a continuum description of ion flow where ionic distributions are described in terms of concentrations. Nonionic interparticle interactions are not considered in this theory as in continuum framework, their impact on the solution is minimal. This theory holds true for dilute flows or flows where channel radius is significantly larger than ion radius. However, for ion channel flows, where channel dimensions and ionic radius are of similar magnitude, nonionic interactions, particularly related to the size of the ions (steric effect), play an important role in defining the selectivity of the channel, concentration distribution of ionic species, and current across the channel, etc. To account for the effect of size of ions, several modifications to PNP equations have been proposed. One such approach is the introduction of Lennard-Jones potential to the energy variational formulation of PNP system. This study focuses on understanding the role of steric effect on flow properties. To discretize the system, Lattice Boltzmann method has been used. The system is defined by modified PNP equations where the steric effect is described by Lennard-Jones potential. In addition, boundary conditions for the complex channel geometry have been treated using immersed boundary method.

Reliability of Poisson-Nernst-Planck Anomalous Models for Impedance Spectroscopy.

We investigate possible connections between two different implementations of the Poisson-Nernst-Planck (PNP) anomalous models used to analyze the electrical response of electrolytic cells. One of them is built in the framework of the fractional calculus and considers integro-differential boundary conditions also formulated by using fractional derivatives; the other one is an extension of the standard PNP model presented by Barsoukov and Macdonald, which can also be related to equivalent circuits containing constant phase elements (CPEs). Both extensions may be related to an anomalous diffusion with subdiffusive characteristics through the electrical conductivity and are able to describe the experimental data presented here. Furthermore, we apply the Bayesian inversion method to extract the parameter of interest in the analytical formulas of impedance. To resolve the corresponding inverse problem, we use the delayed-rejection adaptive-Metropolis algorithm (DRAM) in the context of Markov-chain Monte Carlo (MCMC) algorithms to find the posterior distributions of the parameter and the corresponding confidence intervals.

Mechanistic model of multi-frequency complex conductivity of porous media containing water-wet nonconductive and conductive particles at various water saturations

Electrically conductive particles, such as pyrites, and surface-charge-bearing nonconductive particles, such as clays, are commonly present in water-bearing subsurface formations. Under an external electric field generated by electromagnetic measurement tool, these particles give rise to interfacial polarization (IFP) effects, which causes frequency dispersion of effective conductivity and effective permittivity of the mixture containing such particles. The neglect of IFP effects can lead to inaccurate estimation of petrophysical properties of formations, especially in clay- and pyrite- rich formations. In this paper, we developed a mechanistic model that couples surface-conductance-assisted interfacial polarization (SCAIP) model with perfectly polarized interfacial polarization (PPIP) model to estimate effective conductivity and effective permittivity of homogeneous formations containing both nonconductive and conductive particles at various fluids saturations. The model is developed based on the Poisson-Nernst-Planck (PNP) equations for a dilute solution in a weak electrical field regime to calculate the dipolarizability of the representative volume comprising a single isolated spherical particle in an electrolyte host. Then the effective medium theory is used to determine effective complex conductivity of the whole mixture. The result shows that the conductive particles dominate the frequency dispersion of complex conductivity due to IFP effects compared to nonconductive particles.

The effects of electrostatic correlations on the ionic current rectification in conical nanopores.

Ion-ion electrostatic correlations are recognized to play a significant role in the presence of concentrated multivalent electrolytes. To account for their impact on ionic current rectification phenomenon in conical nanopores, we use the modified continuum Poisson-Nernst-Planck (PNP) equations by Bazant et al. Coupled with the Stokes equations, the effects of the EOF are also included. We thoroughly investigate the dependence of the ionic current rectification ratios as a function of the double layer thickness and the electrostatic correlation length. By considering the electrostatic correlations, the modified PNP model successfully captures the ionic current rectification reversal in nanopores filled with lanthanum chloride LaCl3 . This finding qualitatively agrees with the experimental observations that cannot be explained by the standard PNP model, suggesting that ion-ion electrostatic correlations are responsible for this reversal behavior. The modified PNP model not only can be used to explain the experiments, but also go beyond to provide a design tool for nanopore applications involving multivalent electrolytes.

Numerical simulation and experimental corroboration of galvanic corrosion of mild steel in synthetic concrete pore solution

Corrosion of reinforcing steel in concrete is one of the most prevalent deterioration mechanisms affecting reinforced concrete structures. While there have been significant advances in modeling the initiation stage of corrosion, corrosion kinetic models for predicting the rate of corrosion after depassivation of steel are scarce, and models with experimental corroboration under controlled experimental conditions are virtually nonexistent. Furthermore, the sensitivity of corrosion kinetic models to the uncertainty of their input parameters is not understood.The objective of the present work is to model active corrosion of steel in synthetic solution, experimentally corroborate the modeling approach under controlled conditions, and study the effect of uncertainty of the input parameters on the model predictions. To this end, a two-dimensional finite element method is used to solve the coupled system of Poisson-Nernst-Planck (PNP) equations subjected to electroneutrality constraint. To corroborate the modeling approach, the results of computations are compared against one-dimensional and two-dimensional galvanic corrosion of stainless/carbon steel in dilute and non-dilute NaCl electrolytes as well as two synthetic concrete pore solutions. The modeling parameters, including electrode polarization behaviors and electrolyte properties, are obtained experimentally. Monte Carlo simulations are used to understand the effect of uncertainty of polarization parameters on the predicted corrosion rate.

Bionic Thermoelectric Response with Nanochannels.

Thermosensation, the ability to detect environmental temperature change, is one of most ancient and crucial processes for the survival of all living organisms. Mammals use temperature-sensitive transient receptor potential (thermoTRP) cation channels as thermometers to convert the temperature change into electrical signals that are finally received as action potentials by nerve endings. In this work, we report the bionic thermosensation by solid-state hybrid nanochannels based on the principle of thermally sensitive permselective ion transport. The hybrid nanochannels possess an asymmetric structure, consisting of ultrasmall silica nanochannels (∼2.3 nm in diameter, ∼100 nm in length) supported by large-sized track-etched poly(ethylene terephthalate) conical nanochannels. When the hybrid nanochannels are engineered to separate two electrolyte solutions, the temperature change in one solution can be directly converted into trans-nanochannel diffusion potential, akin to the natural thermosensation process. Two bionic modes, namely, in the absence and presence of a concentration gradient, were studied to imitate the natural thermosensation of thermoTRP ion channels and shark, respectively. In both cases, real-time thermoelectric response was captured with a fast relative response speed (electrical response time versus temperature change time) of higher than 98%, as well as excellent stability and reversibility. Moreover, the nanochannels are highly sensitive to thermal stimulus, showing a sensitivity of 0.71 mV/K comparable to the natural thermosensation. The experimental results coincide well with the theoretical relationship between electrical response and temperature change derived in terms of a quasi-steady-state ion transport model. Finite element simulations based on coupled Poisson-Nernst-Planck (PNP) and Einstein-Stokes equations were also performed, confirming that the sensitive thermoelectric response originates from the highly cationic selectivity of nanochannels.

Multiscale analysis of the effect of surface charge pattern on a nanopore's rectification and selectivity properties: From all-atom model to Poisson-Nernst-Planck.

We report a multiscale modeling study for charged cylindrical nanopores using three modeling levels that include (1) an all-atom explicit-water model studied with molecular dynamics, and reduced models with implicit water containing (2) hard-sphere ions studied with the Local Equilibrium Monte Carlo simulation method (computing ionic correlations accurately), and (3) point ions studied with Poisson-Nernst-Planck theory (mean-field approximation). We show that reduced models are able to reproduce device functions (rectification and selectivity) for a wide variety of charge patterns, that is, reduced models are useful in understanding the mesoscale physics of the device (i.e., how the current is produced). We also analyze the relationship of the reduced implicit-water models with the explicit-water model and show that diffusion coefficients in the reduced models can be used as adjustable parameters with which the results of the explicit- and implicit-water models can be related. We find that the values of the diffusion coefficients are sensitive to the net charge of the pore but are relatively transferable to different voltages and charge patterns with the same total charge.